Question

Find the real or imaginary solutions solutions of the following equation by factoring. y^3-512=0Choose the correct answer below.

Answer 1

B

1) The best way to tackle this question is to think of the difference between two cubes:

$x^3-y^3=(x-y)(x^2+xy+y^2)$

2) So now, let's apply to the binomial we have:

$\begin{gathered} \sqrt[3]{512}=8 \\ y^3-512=(y-8)(y^2+8y+64) \end{gathered}$

So now, let's make use of the factor zero property for the first factor and solve the quadratic using the quadratic formula:

$\begin{gathered} y-8=0,y=8 \\ \\ y_=\frac{-8\pm\sqrt{8^2-4\cdot\:1\cdot\:64}}{2} \\ y_1=\frac{-8+8\sqrt{3}i}{2}=\quad4+4\sqrt{3}i \\ y_2=\frac{-8-8\sqrt{3}i}{2}=\quad-4-4\sqrt{3}i \end{gathered}$

3) Thus, the answer is:

B